Intersection of harmonics and Capelli identities for symmetric pairs

Abstract

We consider a see-saw pair consisting of a Hermitian symmetric pair (G r, Kr) and a compact symmetric pair (Mr, H r), where (Gr, Hr) and (Kr, M r) form a real reductive dual pair in a large symplectic group. In this setting, we get Capelli identities which explicitly represent certain Kc-invariant elements in U(gc) in terms of Hc invariant elements in U(mc). The correspondingHc-invariant elements are called Capelli elements. We also give a decomposition of the intersection of O2n-harmonics and Sp2nharmonics as a module of GL n = O-2n ∩ Sp2n, and construct a basis for the GLn highest weight vectors. This intersection is in the kernel of our Capelli elements. © 2008 The Mathematical Society of Japan.